# NETT: Solving Inverse Problems with Deep Neural Networks

arXiv: https://arxiv.org/abs/1803.00092 · HTML: https://mlsanity.com/paper/1803.00092/

## Key facts

- Cited by 89 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2018.

## Built on by (most cited first, 30 of 89)

- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [End-to-end reconstruction meets data-driven regularization for inverse problems](https://mlsanity.com/paper/2106.03538/) (2021)
- [Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms](https://mlsanity.com/paper/2308.10542/) (2023)
- [Learning the optimal Tikhonov regularizer for inverse problems](https://mlsanity.com/paper/2106.06513/) (2021)
- [What's in a Prior? Learned Proximal Networks for Inverse Problems](https://mlsanity.com/paper/2310.14344/) (2023)
- [Bayesian Imaging With Data-Driven Priors Encoded by Neural Networks: Theory, Methods, and Algorithms](https://mlsanity.com/paper/2103.10182/) (2021)
- [Regularising Inverse Problems with Generative Machine Learning Models](https://mlsanity.com/paper/2107.11191/) (2021)
- [Learning Regularization Parameters of Inverse Problems via Deep Neural Networks](https://mlsanity.com/paper/2104.06594/) (2021)
- [FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Model](https://mlsanity.com/paper/2404.14688/) (2024)
- [Known Operator Learning and Hybrid Machine Learning in Medical Imaging -- A Review of the Past, the Present, and the Future](https://mlsanity.com/paper/2108.04543/) (2021)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation](https://mlsanity.com/paper/2402.01052/) (2024)
- [Globally injective and bijective neural operators](https://mlsanity.com/paper/2306.03982/) (2023)
- [Convergent regularization in inverse problems and linear plug-and-play denoisers](https://mlsanity.com/paper/2307.09441/) (2023)
- [TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems](https://mlsanity.com/paper/2105.12033/) (2021)
- [Blind Inversion using Latent Diffusion Priors](https://mlsanity.com/paper/2407.01027/) (2024)
- [Learning convex regularizers satisfying the variational source condition for inverse problems](https://mlsanity.com/paper/2110.12520/) (2021)
- [Plug-and-Play Posterior Sampling under Mismatched Measurement and Prior Models](https://mlsanity.com/paper/2310.03546/) (2023)
- [Model-corrected learned primal-dual models for fast limited-view photoacoustic tomography](https://mlsanity.com/paper/2304.01963/) (2023)
- [InVAErt networks: a data-driven framework for model synthesis and identifiability analysis](https://mlsanity.com/paper/2307.12586/) (2023)
- [DEALing with Image Reconstruction: Deep Attentive Least Squares](https://mlsanity.com/paper/2502.04079/) (2025)
- [Adversarially learned iterative reconstruction for imaging inverse problems](https://mlsanity.com/paper/2103.16151/) (2021)
- [Reconstructing spectral functions via automatic differentiation](https://mlsanity.com/paper/2111.14760/) (2021)
- [Statistical Learning and Inverse Problems: A Stochastic Gradient Approach](https://mlsanity.com/paper/2209.14967/) (2022)
- [Deep unfolding as iterative regularization for imaging inverse problems](https://mlsanity.com/paper/2211.13452/) (2022)
- [Convergent Data-driven Regularizations for CT Reconstruction](https://mlsanity.com/paper/2212.07786/) (2022)
- [Solving Low-Dose CT Reconstruction via GAN with Local Coherence](https://mlsanity.com/paper/2309.13584/) (2023)

## Abstract (arXiv)

Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel algorithms using deep learning and neural networks for inverse problems appeared. While still in their infancy, these techniques show astonishing performance for applications like low-dose CT or various sparse data problems. However, there are few theoretical results for deep learning in inverse problems. In this paper, we establish a complete convergence analysis for the proposed NETT (Network Tikhonov) approach to inverse problems. NETT considers data consistent solutions having small value of a regularizer defined by a trained neural network. We derive well-posedness results and quantitative error estimates, and propose a possible strategy for training the regularizer. Our theoretical results and framework are different from any previous work using neural networks for solving inverse problems. A possible data driven regularizer is proposed. Numerical results are presented for a tomographic sparse data problem, which demonstrate good performance of NETT even for unknowns of different type from the training data. To derive the convergence and convergence rates results we introduce a new framework based on the absolute Bregman distance generalizing the standard Bregman distance from the convex to the non-convex case.